Stringency of tests for random number generators

by Tso, Chi-wai

Abstract (Summary)

(Uncorrected OCR)
Abstract of thesis entitled
Stringency of tests for random number generators
Submitted by
Tso Chi Wai
for the degree of Master of Philosophy
at The University of Hong Kong
in August 2004
Random number generators (RNGs) are used in statistical simulations. Poor RNGs can cause biased results which are hard to discover. As a preventive measure, statistical tests are developed to check the uniformity and the independence of the numbers generated by an RNG. The usefulness of a test is determined by its ability or power, as it is known, to reject poor RNGs.
Different families of RNGs have different kinds of theoretical properties. It is more appropriate to study powers of a test against different RNG families separately. However, even direct measurement of the power of a test against one single family is difficult. We have therefore developed a measure called stringency, which is an estimation of the power of a test against a specific RNG family.
We treat the randomness of an RNG as its chance of passing all tests suggested in previous studies. For a specific RNG family, a set of canonical RNGs with increasing randomness is chosen. This set of canonical RNGs is called a stringency scale. To gauge the stringency of a test against this family, we apply the test to the canonical RNGs one by one, starting from the least random. The stringency of the test is the number of generators it fails before the first pass. A test with a higher power against the RNG family should be able to achieve higher stringency. In this
study, we established scales for the multiplicative linear congruential generators (MLCGs), the three shift registers generators (SHR3s) and the additive lagged Fibonacci generators (ALFGs) families. These are RNGs commonly used in simulations.
Many tests have parameters. For a test, each set of parameter values specifies a version of the test. The values for the parameters of a test are often chosen without any quantitative analysis. With the stringency scales, the powers of different versions of a test can be compared. This enables us to tune the parameters of a test for maximal power. We have tuned the collision test, the birthday spacing test and the universal test in this way.
We have also compared the powers of the collision test, the birthday spacing test, the universal test and the gorilla test under the condition that all examine the same number of bits. Given a specific number of bits, a test may have many versions which examine all the bits. We first find the most powerful version of each test. Then we compare the most stringent versions of different tests using the scales. The results show that the gorilla test is the most powerful against the MLCG family. The birthday spacing test is the most powerful against the SHR3 and the ALFG families. The collision test is always weaker than the gorilla test. The universal test is the weakest of all tests in all scales.